Arithmetics in numeration systems with negative quadratic base
Kybernetika, Tome 47 (2011) no. 1, pp. 74-92
Voir la notice de l'article provenant de la source Czech Digital Mathematics Library
We consider positional numeration system with negative base $-\beta$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $\beta$ is a quadratic Pisot number. We study a class of roots $\beta>1$ of polynomials $x^2-mx-n$, $m\geq n\geq 1$, and show that in this case the set ${\rm Fin}(-\beta)$ of finite $(-\beta)$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $\beta=\tau=\frac12(1+\sqrt5)$, the golden ratio. For such $\beta$, we determine the exact bound on the number of fractional digits appearing in arithmetical operations. We also show that the set of $(-\tau)$-integers coincides on the positive half-line with the set of $(\tau^2)$-integers.
@article{KYB_2011__47_1_a5,
author = {Mas\'akov\'a, Zuzana and V\'avra, Tom\'a\v{s}},
title = {Arithmetics in numeration systems with negative quadratic base},
journal = {Kybernetika},
pages = {74--92},
publisher = {mathdoc},
volume = {47},
number = {1},
year = {2011},
mrnumber = {2807865},
zbl = {1227.11033},
language = {en},
url = {http://geodesic.mathdoc.fr/item/KYB_2011__47_1_a5/}
}
Masáková, Zuzana; Vávra, Tomáš. Arithmetics in numeration systems with negative quadratic base. Kybernetika, Tome 47 (2011) no. 1, pp. 74-92. http://geodesic.mathdoc.fr/item/KYB_2011__47_1_a5/